Following [1]}, the above property is equivalent to the channel \(p^i(y^i \in \cdot |\omega _0, y^1, u^1, \dots , y^{i-1}, u^{i-1})\)  being a Borel-measurable map from \(\Omega _0 \times \mathbb {Y}^1 \times \mathbb {U}^1 \times \dots \times \mathbb {Y}^{i-1} \times \mathbb {U}^{i-1}\)  to \(\mathcal {P}(\mathbb {Y}^i)\) .
